Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


The Models with Impact Deformations

Discontinuity, Nonlinearity, and Complexity 4(1) (2015) 49--78 | DOI:10.5890/DNC.2015.03.005

M. U. Akhmet; A. Kıvılcım

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

Download Full Text PDF

 

Abstract

We consider mechanisms with impact deformations such that colliding parts are deformable and the Newton’s coefficient of restitution is variable. It is shown how a system with impact deformations can replace the KelvinVoigt viscoelastic model in analysis of a mechanism with contact motion. The suggested impact deformations are compared with the experimental data. By applying deformable surfaces of contacts and non-constant coeffi- cients of restitution, we suppress the chattering in two different mechanical models. We have investigated the existence and stability of periodic solutions in mechanisms with contacts. To actualize the theoretical results, extended examples with simulations are presented.

References

  1. [1]  Brogliato, B. (1999), Nonsmooth Mechanics, Springer-Verlag, London.
  2. [2]  Bishop, S.R. (1994), Impact oscillators, Philosophical Transactions of the Royal Society London A, 347, 347-351.
  3. [3]  Pust, L. and Peterka, F. (2003), Impact oscillator with Hertz's model of contact, Meccanica, 38, 99-114.
  4. [4]  Hedrih, K., Raicevic, V., and Jovic, S. (2011), Phase trajectory portrait of the vibro-impact forced dynamics of two heavy mass particles motions along rough circle,Communications in Nonlinear Science and Numerical Simulation , 16, 4745-4755.
  5. [5]  Blazejczyk-Okolewska, B., Czolczynski, K., and Kapitaniak, T. (2010), Hard versus soft impacts in oscillatory systems modeling, Communications in Nonlinear Science and Numerical Simulation , 15, 1358-1367.
  6. [6]  Luo, G., Zhang, Y., Xie, J., and Zhang, J. (2008), Periodic-impact motions and bifurcations of vibro-impact systems near 1:4 strong resonance point, Communications in Nonlinear Science and Numerical Simulation , 13, 1002-1014.
  7. [7]  Luo, G.W., Lv, H.LX., and Ma, L. (2008), Periodic-impact motions and bifurcation in dynamics of a plastic impact oscillator with a frictional slider, European Journal of Mechanics A/Solids, 27, 1088-1107.
  8. [8]  Burton, R. (1958), Vibration and Impact, Addison-Wesley Publishing Company:New York.
  9. [9]  Pavlovskaia, E. and Wiercigroch, M. (2003), Periodic solution finder for an impact oscillator with a drift,Journal of Sound and Vibration, 267 , 893-911.
  10. [10]  Kapitaniak, T. and Wiercigroch, M. (2000), Dynamics of impact oscillators: An introduction, Chaos, Solitons and Fractals, 11, 2411-2412.
  11. [11]  Awrejcewicz, J. and Lamarque, C.H., (2003) Bifurcation and Chaos in Nonsmooth Mechanical Systems, World Scientific Series on Nonlinear Science, Singapore.
  12. [12]  Fischer-Cripps, A.C.(2000), Introduction to Contact Mechanics, Springer-Verlag, New York.
  13. [13]  Johnson, K.L. (985),Contact Mechanics, Cambridge University Press, United Kingdom.
  14. [14]  Falcon, E., Laroche, C., Fauve, S., and Coste, C. (1998) Behavior of one inelastic ball bouncing repeatedly off the ground, The European Physical Journal B,3, 45-57.
  15. [15]  Vogel, S. and Linz S.J. (2011), Regular and chaotic dynamics in bouncing ball models, International Journal of Bifurcation and Chaos, 21, 869-884.
  16. [16]  Pavlovskaia, E. and Wiercigroch, M. (2001), Modeling of an impact system with a drift, Physical Review E, 64, (056224), 1-9.
  17. [17]  Khan, A.S. and Huang, S. (1995), Continuum Theory of Plasticity, John Wiley and Sons Inc, New York.
  18. [18]  Dill, E.H. (2007), ContinuumMechanics, Elasticity, Plasticity, Viscoelasticity, Taylor and Francis Group, Portland OR.
  19. [19]  Visintin, A. (2006), Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity, Continuum Mechanics and Thermodynamics, 18, 223-252.
  20. [20]  Gomez J.T. and Shukla, A. (2001),Multiple impact penetration of semi-infinite concrete, Journal of Impact Engineering, 25, 965-979.
  21. [21]  Forrestal, M.J., Frew, D.J., Hanchak, S.C., and Brar, N.S. (1996) Penetration of grout and concrete targets with ogivenose steel projectiles, International Journal of Impact Engineering, 18, 465-476.
  22. [22]  Fu, J., Adams, M.J., Reynolds, G.K., Salman, A.D., and Hounslow, M.J. (2004), Impact deformation and rebound of wet granules, Powder Technology, 140, 248-257.
  23. [23]  Biance, A., Chevy, F., Clanet, C., Lagubeau, G., and Quere, D. (2006), On the elasticity of an inertial liquid shock, Journal of Fluid Mechanics, 554, 47-66.
  24. [24]  Lenci, S., Demeio, L., and Petrini, M. (2005), Response scenario and non-smooth features in the nonlinear dynamics of an impacting inverted pendulum, Journal of Computational and Nonlinear Dynamics, 1, 56-64.
  25. [25]  Ibrahim, R.A. (2009) Vibro-impact dynamics modeling, mapping and applications, Springer-Verlag, Berlin Heidelberg.
  26. [26]  Budd, C. and Dux, F. (1994), Chattering and related behavior in impact oscillators, Philosophical Transactions: Physical Sciences and Engineering, 347, 365-389.
  27. [27]  Paparella, F. and Passoni, G. ( 2008), Absence of inelastic collapse for a 1D gas of grains with an internal degree of freedom, Computers and Mathematics with Applications, 55, 218-229.
  28. [28]  Jaeger, H.M. and Nagel, S.R. (1996), Granular solids, liquids, and gases, Reviews of Modern Physics, 68, 1259-1273.
  29. [29]  McNamara, S. and Young,W.R. (1994), Inelastic collapse in two dimension, Physical Reviews E, 50, R28-R31.
  30. [30]  McNamara, S. and Young,W.R. (1992), Inelastic collapse and clumping in one dimensional granular medium, Physical Fluids A, 4, 496-504.
  31. [31]  Marhefka, D.W. and Orin, D.E. (1999) A compliant contact model with nonlinear damping for simulation of robotic systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A:Systems and Humans, 29, 566-572.
  32. [32]  Akhmet, M. (2010), Principle of Discontinuous Dynamical Systems, Springer-Verlag, New York.
  33. [33]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations,World Scientific, Singapore.
  34. [34]  Luo, A.C.J. (2012), Discontinuous Dynamical Systems, Higher Education Press & Springer, Beijing & Berlin.
  35. [35]  Luo, A.C.J. (2006) Singularity and Dynamics on Discontinuous Vector Fields, Volume 3 (Monograph Series on Nonlinear Science and Complexity) Elsevier, Netherlands.
  36. [36]  Akalin, E. and Akhmet, M.U. (2005), The principles of B-smooth discontinuous flows, Computers & Mathematics with Applications , 49, 981-995.
  37. [37]  Thomson,W.T. (1988), Theory of Vibration with Applications, Prentice-Hall, New Jersey.
  38. [38]  Taylor, J.R.A. and Patek, S.N. (2010) Ritualized fighting and biological armor: the impact mechanics of the mantis shrimp's telson, The Journal of Experimental Biology, 213, 3496-3504.
  39. [39]  Zheltukhin, S. and Lui, R. (2011), One-dimendional viscoelastic cell motility models, Mathematical Biosciences, 229, 30-40.
  40. [40]  Argatov, I.I. (2013), Mathematical modeling of linear viscoleastic impact: Application to drop impact testing of articular cartilage, Tribology International, 63, 213-225.
  41. [41]  Nagurka, M. and Huang, S. (2006), A mass-spring-damper model of a bouncing ball, International Journal of Engineering Education, 22, 393-401.
  42. [42]  Andronow, A.A. and Chaikin, C.E. (1949), Theory of Oscillations, Princeton University Press, Princeton.
  43. [43]  Babitsky, V.I. (1998), Theory of Vibro-Impact System and Applications, Springer, Berlin.
  44. [44]  Luo, A.C.J. and Guo, Y. (2013), Vibro-impact Dynamics, John Wiley & Sons, Ltd, United Kingdom.
  45. [45]  Thomsen, J.J. and Fidlin, A. (2008), Near-elastic vibro-impact analysis by discontinuous transformations and averaging, Journal of Sound and Vibration, 311, 386-407.
  46. [46]  McNamara, S. and Falcon, E. (2005), Simulations of vibrated granular medium with impact velocity dependent restitution coefficient, Physical Review E, 71, (031302), 1-6.
  47. [47]  di Bernardo, M. , Budd, C.J., Champneys, A.R., and Kowalczyk, P. (2008), Piecewise-smooth Dynamical Systems Theory and Applications, Springer-Verlag,London.
  48. [48]  Luo, A.C.J. (2010), Nonlinear Deformable-body Dynamics, Higher Educaiton Press & Springer, Beijing & Berlin.
  49. [49]  Luo, G and Xie, J. (2001), Bifurcation and chaos in a system with impacts, Physica D, 148, 183-200.
  50. [50]  Akhmet,M. and Turan, M. (2013), Bifurcation of discontinuous limit cycles of the Van der Pol equation, Mathematics and Computers in Simulation, 95, 39-54.
  51. [51]  Demeio, L. and Lenci, S. (2006), Asymptotic analysis of chattering oscillations for an impacting inverted pendulum, Journal of Applied Mathematics and Mechanics, 59, 419-434.
  52. [52]  Jackson, R.L., Green, I., and Marghitu, D.B. (2010), Predicting the coefficient of restitution of impacting elasticperfectly plastic spheres, Nonlinear Dynamics, 60, 217-229.
  53. [53]  Wu, C. Y., Li, L.Y., and Thornton, C. (2005), Energy dissipation during normal impact of elastic and elastic-plastic spheres, International Journal of Impact Engineering , 32, 593-604.
  54. [54]  Guisepponi, S., Marchesoni, F., and Borromeo, M. (2005), Randomness in the bouncing ball dynamics, Physica A, 351, 142-158.
  55. [55]  Wagg, D.J. and Bishop, S.R. (2001), Chatter, sticking and chaotic impacting motion in a two degree of freedom impact oscillator, International Journal of Bifurcation and Chaos, 11, 57-71.
  56. [56]  Luck, J.M. and Metha, A. (1993), Bouncing ball with a finite restitution: Chattering, locking, and chaos, Physical Reviev E, 48, 3988-3997.
  57. [57]  Luo, A.C.J. and O'Connor, D. (2009), Periodic motions and chaos with impacting chatter and stick in a gear transmission system, International Journal of Bifurcation and Chaos, 19, 1975-1994.
  58. [58]  Luo, A.C.J. and O'Connor, D. (2009), Mechanisms of impacting chatter with stick in a gear transmission system, International Journal of Bifurcation and Chaos, 19, 2093-2105.
  59. [59]  Nagaev, R.F. (1985), Mechanical processes with repeated attenuated impacts, World Scientific Publishing Co, Singapore.
  60. [60]  Akhmet, M. and Turan, M. (2010), Bifurcation in a 3D hybrid system, Communications in Applied Analysis, 14, 311-324.
  61. [61]  Stronge,W.J. (2000), Impact Mechanics, Cambridge University Press, Cambridge.
  62. [62]  Tabor, D.(1948), A simple theory of static and dynamic hardness, Proceedings of the Royal Society London A, 192, 247-274.
  63. [63]  Hunt, K.H. and Crossley, F.R.E. (1975), Coefficient of restitution interpreted as damping in vibroimpact, ASME Journal of Applied Mechanics, 42, 440-445 .
  64. [64]  Ghayesh, M.H. (2012), Nonlinear dynamic response of a simply- supported Kelvin-Voigt viscoelastic beam, Nonlinear Analysis, Real World Applications, 13, 1319-1333.
  65. [65]  Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.
  66. [66]  Robinson, C. (1995), Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC, Boca Raton, Ann Arbor, London, Tokyo.
  67. [67]  Tung, P.C, Shaw SW. (1988), The dynamics of an impact print hammer, Journal of Vibration, Acoustics, Stress, and Reliability in Design, 110, 193-200.